A differential equation with discontinuous right-hand side would appear
in general to be unduly sensitive to initial conditions, or to the
presence of perturbation or error terms. For this reason, one might
doubt that discontinuous stabilizing feedback would be effective in the
presence of such terms. Yet a large class of such feedbacks constructed
through ``proximal aiming'' has desirable features in this regard. We
describe this precisely, as well as another approach which limits the
discontinuities to a small tractable set.

This paper considers control design for distributed systems, where the
controller is to adopt and preserve the distributed spatial structure
of the system. Specifically systems that are inhomogeneous with
respect to the spatial variable, and may have imposed boundary
conditions, are considered. Operator theoretic tools for working with
these systems are developed, and lead to convex conditions for analysis
and synthesis with respect to the l2-induced norm. Our general
motivation is to develop tools for dealing with physical systems that
are inherently distributed and are to be controlled using a distributed
strategy, as well as the increasing number of systems that are
distributed because they are comprised of distributed interacting
subsystems which have only local control available, but nonetheless
require global performance objectives to be achieved.

One can identify at least three eras in linear control design in this
century: classical, modern, and postmodern. Postmodern refers to the
multivariable design approach developed mostly in the 1980s starting
from Zames' famous 1981 paper.

This talk will begin with a historical review from the viewpoint of a
volume in preparation of 25 seminal control papers of this century.
Then some case studies will be presented, including telerobotics, large
flexible space structures, and web-handling systems.

The human body in health and disease exhibits a host of complex
rhythms. Most are normal, but others may indicate serious illness or
even lead to sudden death. I will review efforts underway to
characterize complex body rhythms, and to understand their underlying
mechanisms. If the origin of the rhythms are understood (or perhaps
even if they are not) it may be possible to control the rhythms to
establish normal functioning or to prevent the onset of more dangerous
rhythms. There is a well-established medical device industry that
successfully controls abnormal cardiac and neural rhythms, but there is
often minimal theoretical analysis of underlying mechanisms. However,
various strategies to control cardiac and neural heart rhythms have
been proposed that are based on nonlinear mathematical models of the
abnormal rhythms.

We consider the receding horizon optimal control problem with terminal
cost chosen as control Liapunov function and analyze the asymptotic
behavior of solutions for regulator as well as disturbance attenuation
problems.

The affine connection formalism is a powerful one in the
study of control theory for a large class of mechanical systems,
namely those which are Lagrangian with Lagrangian equal to kinetic
minus potential energy. The tools of affine differential geometry
enable an investigation of some basic questions surrounding control
theory for such systems, and topics such as controllability, optimal
control, and feedback equivalence will be discussed.

A practical and important class of control problems concern systems
modelled by partial differential equations. In general, it is
necessary to use a numerical approximation to simulate the response
of the system and to compute controllers for the system. A scheme
that yields good results when used for simulation may be inappropriate
for use in controller design. For instance, a scheme for which the
simulation results converge quickly may have a corresponding
controller sequence that converges very slowly or not at all. These
issues will be discussed with respect to several common applications.
Criteria appropriate for evaluation of an approximation scheme will
be discussed.

Over the last decade or so several different methods have been
developed to solve the H-infinity optimal controller design problem for
infinite dimensional systems. One of these methods is called the
``skew Toeplitz'' approach: it was shown that the singular values and
vectors of an infinite dimensional ``Hankel+Toeplitz'' operator (which
determines the optimal H-infinity controller) are computed from
invertibility conditions of a skew Toeplitz operator. Under certain
mild assumptions these invertibility conditions can be reduced to
finitely many interpolation conditions. Thus the H-infinity
controller can be derived by solving a finite set of linear equations.
In this presentation we will give a simplified formula for H-infinity
controllers for single input single output plants for which inner outer
factorizations can be done numerically. The special structure of this
controller expression will be exploited to derive finite dimensional
(rational) approximations of the optimal controller. Direct
approximation of the controller will be compared to indirect
approximations. Numerical examples will be given from time delay
systems.

This paper is concerned with forced propagation of constant velocity
waves in an elastic beam modelled by a nonlinear partial differential
equation of type corresponding to the full von Karman model in plate
theory. An elastic beam, supported on a flat, inelastic surface and
subject to a vertical gravitational force, is subjected to a
compressive force, tending to create buckling away from the surface. A
heavy moving weight forces the buckled state to move at constant
velocity c. Using analytical and computational studies we examine
the properties of the resulting travelling wave as they depend on the
various parameters present in the problem and on the propagation
velocity c. These properties include the geometric form of the
travelling wave and its stability aspects.

In problems of stabilization and optimal control, it is generally the
case that effective continuous feedback controls do not exist. In
recent years, the tools of nonsmooth analysis have been employed in
order to produce discontinuous feedbacks. In this talk,the application
of these techniques to a problem of state constrained optimal control
will be described.